Liouville Brownian motion at criticality

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Liouville Brownian motion at criticality

Rémi Rhodes, Vincent Vargas

To cite this version:

Rémi Rhodes, Vincent Vargas. Liouville Brownian motion at criticality. Potential Analysis, Springer Verlag, 2015, 43 (2), pp.149-197. �hal-00914387�

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Liouville Brownian motion at criticality

R´emi Rhodes ^∗ Vincent Vargas ^†

Abstract

In this paper, we construct the Brownian motion of Liouville Quantum gravity when the underlying conformal field theory has a c = 1 central charge. Liouville quantum gravity with c= 1 corresponds to two-dimensional string theory and is the conjectural scaling limit of large planar maps weighted with aO(n= 2) loop model or aQ= 4-state Potts model embedded in a two dimensional surface in a conformal manner.

Following [27], we start by constructing the critical LBM from one fixed point x∈R² (or x∈S²), which amounts to changing the speed of a standard planar Brownian motion depending on the local behaviour of the critical Liouville measure M^′(dx) = −X(x)e²^X⁽^x⁾dx (where X is a Gaussian Free Field, say onS²). Extending this construction simultaneously to all points in R² requires a fine analysis of the potential properties of the measure M^′. This allows us to construct a strong Markov process with continuous sample paths living on the support of M^′, namely a dense set of Hausdorff dimension 0. We finally construct the Liouville semigroup, resolvent, Green function, heat kernel and Dirichlet form of (critical) Liouville quantum gravity with ac= 1 central charge.

In passing, we extend to quite a general setting the construction of the critical Gaussian multiplicative chaos that was initiated in [20,21].

Key words or phrases:Gaussian multiplicative chaos, critical Liouville quantum gravity, Brownian motion, heat kernel, potential theory.

MSC 2000 subject classifications: 60J65, 81T40, 60J55, 60J60, 60J80, 60J70, 60K40

1 Introduction

1.1 Physics motivations

Liouville field theory is a two dimensional conformal field theory which plays an important part in two dimensional models of quantum gravity. Euclidean quantum gravity is an attempt to quan- tize general relativity based on Feynman’s functional integral and on the Einstein-Hilbert action principle. One integrates over all Riemannian metrics on ad-dimensional manifold Σ.

General relativity is a reparametrization invariant theory which can be formulated with no reference to coordinates at all and this diffeomorphism invariance is a central issue in the quantum theory. The main motivation for considering 2d quantum gravity comes from the fact that the Einstein-Hilbert action becomes trivial in 2das it reduces to a topological term and the cosmological constant coupled to the volume of space-time. This is the famous representation of the functional integral over geometries as a Liouville field theory by Polyakov [52] (see also [52,40,15]).

More precisely, one couples a Conformal Field Theory (CFT) (or more generally a matter field or quantum field theory) to gravity via any reparametrization invariant action for conformal matter fields with central charge c . A famous example is the coupling of c free scalar matter fields to gravity, which can also be interpreted as an embedding of Σ into a c-dimensional Euclidian space, thus leading to an interpretation of such a specific theory of 2d-Liouville Quantum Gravity as a bosonic string theory in cdimensions [52].

It is shown in [52, 40, 15]) that the reparametrization invariant action of the CFT can be factorized as a tensor product where the metric is independent of the CFT and roughly takes on the form [52,40,15] (we consider an Euclidean background metric for simplicity):

g(x) =e^γX(x)dx², (1.1)

where the fluctuations of the field X are governed by the Liouville action and the parameter γ is related to the central charge of the CFT via the famous result in [40] (forc61)

γ = 1

√6(√

25−c−√

1−c). (1.2)

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Therefore γ ∈[0,2]. When the cosmological constant in the Liouville action is set to 0, one talks about critical2d-Liouville Quantum Gravity and the Liouville action turns the field X in (1.1) into a Free Field, with appropriate boundary conditions. The reader is referred to [14, 15,22,18, 19,26,29,31,38,40,48,52] for more insights on 2d-Liouville quantum gravity. For these reasons and though it may be an interesting field in its own right, Liouville field theory (governing the metric in 2d-Liouville quantum gravity) is an important object in theoretical physics.

In the subcritical case γ <2, c <1, the geometry of the metric tensor (1.1) is mathematically investigated in [22,27,28,55,57] and the famous KPZ scaling relations [40] are rigorously proved in [4,22,56] in a geometrical framework (see also [16] for a non rigorous heat kernel derivation).

This paper focuses on the coupling of a CFT with ac= 1 central charge to gravity or equivalently 2d string theory, i.e. the coupling of one free scalar matter field to gravity: in this case, one has γ = 2 by relation (1.2). For an excellent review on 2dstring theory, we refer to Klebanov’s lecture notes [38]. As expressed by Klebanov in [38]: ”Two-dimensional string theory is the kind of toy model which possesses a remarkably simple structure but at the same time incorporates some of the physics of string theories embedded in higher dimensions”. Among theγ62 theories, the case γ = 2 probably possesses the richest structure, inherited from its specific status of phase transition.

For instance, the construction of the volume form, denoted byM^′, associated to the metric tensor (1.1) is investigated in [20,21] where it is proved that it takes on the unusual form:

M^′(dx) =−X(x)e^2X(x)dx, (1.3)

which also coincides with a proper renormalization ofe^2X(x)dx. Another specific point, the γ = 2 theory possesses non trivial conformally invariant gravitationally dressed vertex operators, the so- calledtachyonic fields: the reader is referred for instance to [38] for more physics insights and to [45] for their rigorous construction and a field derivation of the KPZ formula of these operators.

In this paper, we will complete this picture by constructing the Brownian motion (called critical Liouville Brownian motion, critical LBM for short), semi-group, resolvent, Dirichlet form, Green function and heat kernel of the metric tensor (1.1) with γ = 2. The question of constructing the associated distance is discussed below.

We further point out that Liouville quantum gravity is conjecturally related to discrete and continuum random surfaces. Roughly speaking, when one takes a random two-dimensional manifold and conformally maps it to a disk or the sphere, the image of the metric tensor of the manifold is a metric tensor on the disk or sphere that should correspond to an exponential of a log-correlated Gaussian random variable (some form of the GFF). The reader may consult [22,60], which contain an extensive overview of the physics literature for a c < 1 central charge together with related conjectures and [20] in the case c = 1. Discrete critical statistical physical models having c = 1 then include one-dimensional matrix models (also called “matrix quantum mechanics” (MQM)) [9, 29, 30, 32, 33, 37, 39, 50, 51, 61], the so-called O(n) loop model on a random planar lattice forn= 2 [41,42,43,44], and theQ-state Potts model on a random lattice for Q= 4 [8,13,24].

The critical LBM is therefore conjectured to be the scaling limit of random walks on large planar maps weighted with a O(n = 2) loop model or a Q = 4-state Potts model, which are embedded in a two dimensional surface in a conformal manner. For an introduction to the above mentioned 2d-statistical models, see, e.g., [49]. We further mention [12,17] for recent advances on this topic in the context of pure gravity, i.e. with no coupling with a CFT.

To complete this overview of physics literature, we point out that the notions of diffusions or heat kernel are at the core of physics literature about Liouville quantum gravity (see [2,3,10,11, 14,16,62] for instance): for more on this, see the subsection on the associated distance.

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1.2 Strategy and results

Basically, our approach of the metric tensor e^2X(x)dx² (where X is a Free Field) relies on the construction of the associated Brownian motion, called the critical LBMB. Standard results of 2d- Riemannian geometry tell us that the law of this Brownian motion is a time change of a standard planar Brownian motionB (starting from 0):

Bt^x=x+B_hB^x_i_t (1.4)

where the quadratic variations hB^xi are formally given by:

hB^xit=F^′(x, t)⁻¹, F^′(x, t) = Z t

0

e^2X(x+B^r⁾dr. (1.5)

Put in other words, we should integrate the weighte^2X(x) along the paths of the Brownian motion x+Bto construct a mappingt7→F^′(x, t). The inverse of this mapping corresponds to the quadratic variations ofB^x. Of course, because of the irregularity of the fieldX, giving sense to (1.5) is not straightforward and one has to apply a renormalization procedure: one has to apply a cutoff to the field X (a procedure that smoothes up the field X) and pass to the limit as the cutoff is removed. The procedure is rather standard in this context. Roughly speaking, one introduces an approximating field X_ǫ where the parameter ǫstands for the extent to which one has regularized the field X (we haveX_ǫ→X asǫgoes to 0). One then defines

F^ǫ(x, t) = Z t

0

e^2X^ǫ^(x+B^r⁾dr (1.6)

and one looks for a suitable deterministic renormalization a(ǫ) such that the family a(ǫ)F^ǫ(x, t) converges towards a non trivial object as ǫ → 0. In the subcritical case γ < 2, the situation is rather well understood as the familya(ǫ) roughly corresponds to

a(ǫ)≃exp(−γ²

2 E[X_ǫ(x)²])

(the dependence of the point x is usually fictive and may easily get rid of) in such a way that a(ǫ)

Z t

0

e^γX^ǫ^(x+B^r⁾dr (1.7)

converges towards a non trivial limit. We will call this renormalization procedure standard: it has been successfully applied to construct random measures of the form e^γX(x)dx [34, 58, 59] (the reader may consult [55] for an overview on Gaussian multiplicative chaos theory). Though the choice of the cutoff does not affect the nature of the limiting object, a proper choice of the cutoff turns the expression (1.7) into a martingale. This is convenient to handle the convergence of this object.

At criticality (γ = 2), the situation is conceptually more involved. It is known [20] that the standard renormalization procedure of the volume form yields a trivial object. Logarithmic corrections in the choice of the familya(ǫ) are necessary (see [20,21]) and the limiting measure that we get is the same as that corresponding to a metric tensor of the form −X(x)e^2X^(x). Observe that it is not straightforward to see at first sight that such metric tensors coincide, or even are

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positive. This subtlety is at the origin of some misunderstandings in the physics literature where the two forms of the tachyon fielde^2X(x) and −X(x)e^2X^(x) appear without making perfectly clear that they coincide.

The case of the LBM at criticality obeys this rule too. We will prove that non standard logarithmic corrections are necessary to make the change of time F^ǫ converge and they produce the same limiting change of times as that corresponding to a metric tensor of the form −X(x)e^2X^(x). This summarizes the almost sure convergence of F^ǫ for one given fixed point x∈ R². Yet, if one wishes to define a proper Markov process, one has to go one step further and establish that, almost surely, F^ǫ(x,·) converges simultaneously for all possible starting points x ∈ R²: the place of the

”almost sure” is important and gives rise to difficulties that are conceptually far different from the construction from one given fixed point. In [27], it is noticed that this simultaneous convergence is possible as soon as the volume formM_γ(dx) associated to the metric tensor (1.1) is regular enough so as to make the mapping

x7→

Z

R²

ln₊ 1

|x−y|M_γ(dx)

continuous. When γ <2, multifractal analysis shows that the measure M_γ possesses a power law decay of the size of balls and this is enough to ensure the continuity of this mapping. In the critical case γ = 2, the situation is more complicated because the measure M^′ is rather wild: for instance the Hausdorff dimension of its support is zero [5]. Furthermore, the decay of the size of balls investigated in [5] shows that continuity and even finiteness of the mapping

x7→

Z

R²

ln+

1

|x−y|M^′(dx) (1.8)

is unlikely to hold on the whole of R². Yet we will show that we can have a rather satisfactory control of the size of balls for all xbelonging to a set of full M^′-measure, call itS:

∀x∈S, sup

r∈]0,1]

M^′(B(x, r))(−lnr)^p<+∞

for some p large enough. In particular this estimate shows that the expression (1.8) is finite for every point x ∈ S. Also, this estimate answers a question raised in [5] (a similar estimate was proved by the same authors [6] in the related case of the discrete multiplicative cascades). We will thus construct the change of time F^′(x,·) simultaneously for all x∈S. What happens on the complement of S does not matter that much since it is a set with null M^′ measure. Yet we will extend the change of timeF^′(x,·) to the whole of R².

Once F^′ is constructed, potential theory [25] tells us that the LBM at criticality is a strong Markov process, which preserves the critical measure M^′. We will then define the semigroup, resolvent, Green function and heat kernel associated to the LBM at criticality.

We stress that, once all the pieces of the puzzle are glued together, this LBM at criticality appears as a rather weird mathematical object. It is a Markov process with continuous sample paths living on a very thin set, which is dense inR² and has Hausdorff dimension 0. And in spite of this rather wild structure, the LBM at criticality is regular enough to possess a (weak form of) heat kernel. Beyond the possible applications in physics, we feel that the study of such an object is a fundamental and challenging mathematical problem, which is far from being settled in this paper.

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1.3 Discussion about the associated distance

An important question related to this work is the existence of a distance associated to the metric tensor (1.1) forγ ∈[0,2]. The physics literature contains several suggestions to handle this problem, part of which are discussed along the forthcoming speculative lines. Also since we will base part of our discussion on the results established in [57], we will focus on the non critical case γ < 2 though a similar discussion should hold in the critical case. In this context, we denote F(x, t) = Rt

0e^γX(x+B^r⁾dr the associated additive functional along the Brownian paths. We know that for all t >0 there exists a Liouville heat kernel p^X_t (x, y). Many papers in the physics literature have argued that the Liouville heat kernel should have the following representation which is classical in the context of Riemannian geometry

p^X_t (x, y) ∼

d(x,y)→0

e⁻^d(x,y)²^/t

t (1.9)

whered(x, y) is the associated distance, i.e. the ”Riemannian distance” defined by (1.1). From the representation (1.9), physicists [2,3,62] have derived many fractal and geometrical properties of Liouville quantum gravity. In particular, the paper [62] established a intriguing formula for the dimensiond_H of Liouville quantum gravity which can be defined by the heuristic

M_γ({y; d(x, y)6r})∼r^d^H

where M_γ is the associated volume form. Note that the meaning of the above definition is not obvious sinceMγ is a multifractal random measure.

Along the same lines, a recent physics paper [16] establishes an interesting heat kernel derivation of the KPZ formula. The idea behind the paper is that, if relation (1.9) holds, then one can extract the metric from the heat kernel by using the Mellin-Barnes transform given by

Z _∞

0

t^s⁻¹p^X_t (x, y)dt.

Indeed, a standard computation gives the following equivalent fors∈]0,1[

Z _∞

0

t^s⁻¹e⁻^d(x,y)²^/t

t dt ∼

d(x,y)→0

Cs

d(x, y)²⁽¹⁻^s) (1.10) whereCs is some positive constant. Equivalently, we would have

d(x, y)∼ 1

(R_∞

0 t^s⁻¹p^X_t (x, y)dt)^2(1−s)¹ .

Essentially, the authors of [16] prove the following Mellin-Barnes version of the KPZ formula. LetK be some compact set such that theq-Hausdorff measureH^q(K) is non trivial, i.e. 0<H^q(K)<∞. Then the authors claim that one should roughly have

Z

K

Z

K

Z _∞

0

t⁻^q^¯p^X_t (x, y)dt

e^qγX(x)^¯ ⁻^{( ¯}^qγ)2² ^E[X^(x)²^]H^q(dx)e^qγX(y)^¯ ⁻^{( ¯}^qγ)2² ^E[X(y)²^]H^q(dy)<∞ (1.11) where

e^qγX(x)^¯ ⁻^{( ¯}^qγ)2² ^E[X(x)²^]H^q(dx) := lim

ǫ→0e^qγX^¯ ^ǫ^(x)⁻^{( ¯}^qγ)2² ^E[X^ǫ^(x)²^]H^q(dx)

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and ¯q is related to q by the KPZ relation (2 +^γ₂²)¯q−^γ₂²q¯²=q. Thanks to the relation (1.10), the authors claim that this yields a geometrical version of the KPZ equation which does not rely on the Euclidean metric. Recall that the rigorous geometrical derivations of KPZ in [22,56] rely on the measureM_γ and imply working with Euclidean balls.

All these physicist derivations are a bit puzzling. It seems that the first question that can be settled mathematically is inequality (1.11). Indeed, by the results of [57], one has a very nice representation of a ”natural version” of the Mellin -Barnes transform in terms of Brownian bridges;

more specifically, one has Z _∞

0

t^s⁻¹p^X_t (x, y)dt= Z _∞

0 E^x[F(x, t)^s⁻¹|B_t=y]e⁻^|y−x|

2 2t

2πt dt.

Note that we mention a ”natural version” because it is not obvious to make sense of the measure p^X_t (x, y)dt. If inequality (1.11) is true, it could be the case that, rather than (1.9), the heat kernel has the following behaviour compatible with the KPZ relation derived in [56]

p^X_t (x, y) ∼

|y−x|→0

e⁻^M^γ^(B(x,^|^y⁻^x^|^))/t

t (1.12)

where B(x,|y−x|) is the Euclidean ball of center x and radius |y−x|. In all cases, we believe that these questions are interesting and we hope that this informal discussion will trigger further investigations among mathematicians.

Acknowledgements

The authors wish to thank Antti Kupiainen, Miika Nikula and Christian Webb for many very interesting discussions which have helped a lot in understanding the specificity of the critical case and Fran¸cois David who always take the time to answer their questions with patience and kindness.

2 Setup

In this section, we draw up the framework to construct the Liouville Brownian motion at criticality on the whole plane R². Other geometries are possible and discussed at the end of the paper.

2.1 Notations

In what follows, we will consider Brownian motionsBor ¯BonR²(or other geometries) independent of the underlying Free Field. We will denote byE^Y orP^Y expectations and probability with respect to a fieldY. For instance, E^X or P^X (resp. E^B orP^B) stand for expectation and probability with respect to the log-correlated field X (resp. the Brownian motion B). For d>1, we consider the space C(R₊,R^d) of continuous functions from R₊ intoR^d equipped with the topology of uniform convergence over compact subsets ofR₊.

2.2 Representation of a log-correlated field

In this section we introduce the log-correlated Gaussian fields X on R² that we will work with throughout this paper. One may consider other geometries as well, like the sphereS² or the torus

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T² (in which case an adaptation of the setup and proofs is straightforward). We will represent them via a white noise decomposition.

We consider a family of centered Gaussian processes ((X_ǫ(x))_x_∈_R²)_ǫ>0with covariance structure given, for 1>ǫ > ǫ^′>0, by:

K_ǫ(x, y) =E[X_ǫ(x)X_ǫ^′(y)] = Z ¹_ǫ

1

k(u, x, y)

u du (2.1)

for some family (k(u,·,·))_u>1 of covariance kernels satisfying:

A.1 kis nonnegative, continuous.

A.2 kis locally Lipschitz on the diagonal, i.e. ∀R >0, ∃C_R>0,∀|x|6R,∀u>1,∀y∈R²

|k(u, x, x)−k(u, x, y)|6C_Ru|x−y| A.3 ksatisfies the integrability condition: for each compact set S,

sup

x∈S,y∈R²

Z _∞

1

|x−y|

k(u, x, y)

u du <+∞. A.4 the mapping H_ǫ(x) =R¹_ǫ

1 k(u,x,x)

u du−ln¹_ǫ converges pointwise as ǫ→ 0 and for all compact setK,

sup

x,y∈K

sup

ǫ∈]0,1]

|H_ǫ(x)−H_ǫ(y)|

|x−y| <+∞. A.5 for each compact set K, there exists a constant C_K >0 such that

k(u, x, y)>k(u, x, x)(1−C_Ku^1/2|x−y|^1/2)₊ for all x∈K andy ∈R². for all u>1,x∈K and y∈R².

Such a construction of Gaussian processes is carried out in [1,54] in the translation invariant case.

Furthermore, [A.2] implies the following relation that we will use throughout the paper: for each compact set S, there exists a constant cS > 0 (only depending on k) such that for all y ∈ S, ǫ∈(0,1] andw∈B(y, ǫ), we have

ln1

ǫ −c_S 6K_ǫ(y, w)6 ln1

ǫ +c_S. (2.2)

We denote by Fǫ the sigma algebra generated by{X_u(x);ǫ6u, x∈R²}. 2.3 Examples

We explain first a Fourier white noise decomposition of log-correlated translation invariant fields as this description appears rather naturally in physics. Consider a nonnegative even function ϕ defined onR² such that lim_|_u_|→∞|u|²ϕ(u) = 1. We consider the kernel

K(x, y) = Z

R²

eⁱ^h^u,x⁻^yⁱϕ(u)du. (2.3)

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We consider the following cut-off approximations Kǫ(x, y) = 1

2π Z

B(0,ǫ⁻¹)

eⁱ^h^u,x⁻^yⁱϕ(u)du. (2.4) The kernel K can be seen as the prototype of kernels of log-type in dimension 2. It has obvious counterparts in any dimension. The cut-off approximation is quite natural, rather usual in physics (sometimes called the ultraviolet cut-off) and has well known analogues on compact manifolds (in terms of series expansion along eigenvalues of the Laplacian for instance). If X has covariance given by (2.3), then X has the following representation

X(x) = Z

R²

eⁱ^h^u.xⁱp

ϕ(u)(W₁(du) +iW₂(du))

whereW₁(du) andW₂(du) are independent Gaussian distributions. The distributions W₁(du) and W2(du) are functions of the field X (since they are the real and imaginary parts of the Fourier transform ofX). The law ofW₁(du) isW(du)+W(−du) whereW is a standard white noise and the law ofW₂(du) isWf(du)−Wf(−du) whereWf is also standard white noise. One can then consider the following family with covariance (2.4) and which fits into our framework as it corresponds to adding independent fields

X_ǫ(x) = Z

B(0,ǫ⁻¹)

eⁱ^h^u.xⁱp

ϕ(u)(W₁(du) +iW₂(du))

Notice also that the approximations X_ǫ are functions of the original field X since W₁(du) and W₂(du) are functions of the fieldX.

Notice that K_ǫ can be rewritten as (S stands for the unit sphere and ds for the uniform probability measure onS)

K_ǫ(x, y) = Z ¹_ǫ

0

k(r, x, y)

r dr, withk(r, x, y) =r² Z

S

ϕ(rs) cos(rhx−y, si)ds.

Let us simplify a bit the discussion by assuming that ϕ is isotropic. In that case, it is plain to check that assumptions [A.1-5] are satisfied (in the slightly extended context of integration over [0,¹_ǫ] instead of [1,¹_ǫ] but this is harmless as K₁(x, y) is here a very regular Gaussian kernel).

Example 1. Massive Free Field (MFF). The whole plane MFF is a centered Gaussian distribution with covariance kernel given by the Green function of the operator 2π(m²− △)⁻¹ on R², i.e. by:

∀x, y∈R², G_m(x, y) = Z _∞

0

e⁻^m

2

2 u−^|x−y|2u²du

2u. (2.5)

The real m > 0 is called the mass. This kernel is of σ-positive type in the sense of Kahane [34]

since we integrate a continuous function of positive type with respect to a positive measure. It is furthermore a star-scale invariant kernel (see [1, 54]): it can be rewritten as

G_m(x, y) = Z +∞

1

k_m(u(x−y))

u du. (2.6)

for some continuous covariance kernel k_m(z) = ¹₂R_∞

0 e⁻^m

2

2v|z|²−^v₂ dv and therefore satisfies the assumptions [A.1-5].

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One may also choose the Fourier white noise (2.4) decomposition with ϕ(u) = ¹

|u|²+m² or the semigroup covariance structure

E[X_ǫ(x)X_ǫ^′(y)] =π Z _∞

max(ǫ,ǫ^′)²

p(u, x, y)e⁻^m

2 2 udu,

which also satisfies assumptions [A.1-5] (modulo a change of variable u = 1/v² in the above expression: see [21, section D]).

Example 2. Gaussian Free Field (GFF).Consider a bounded open domainDofR². Formally, a GFF on D is a Gaussian distribution with covariance kernel given by the Green function of the Laplacian on D with prescribed boundary conditions. We describe here the case of Dirichlet boundary conditions. The Green function is then given by the formula:

GD(x, y) =π Z _∞

0

pD(t, x, y)dt (2.7)

where pD is the (sub-Markovian) semi-group of a Brownian motion B killed upon touching the boundary of D, namely for a Borel setA⊂D

Z

A

p_D(t, x, y)dy =P^x(B_t∈A, T_D > t)

with T_D =inf{t>0, B_t 6∈D}. The most direct way to construct a cut-off family of the GFF on D is then to consider a white noise W distributed onD×R₊ and to define:

X(x) =√ π

Z

D×R₊

pD(s

2, x, y)W(dy, ds).

One can check that E[X(x)X(x^′)] = πR_∞

0 p_D(s, x, x^′)ds = G_D(x, x^′). The corresponding cut-off approximations are given by:

Xǫ(x) =√ π

Z

D×[ǫ²,∞[

pD(s

2, x, y)W(dy, ds).

They have the following covariance structure E[X_ǫ(x)X_ǫ^′(y)] =π

Z _∞

max(ǫ,ǫ^′)²

p_D(u, x, y)du, (2.8)

which also satisfies assumptions [A.1-5] (on every subdomain ofDand modulo a change of variable u= 1/v² in the above expression: see also [21, section D]).

For some technical reasons, we will sometimes also consider either of the following assumptions:

A.6 k(v, x, y) = 0 for|x−y|>Dv⁻¹(1 + 2 lnv)^α for some constantsD, α >0, A.6’ (k(v, x, y))_v is the family of kernels presented in examples 1or 2.

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2.4 Regularized Riemannian geometry

We would like to consider a Riemannian metric tensor on R² (using conventional notations in Riemannian geometry) of the typee^2X^ǫ^(x)dx², wheredx² stands for the standard Euclidean metric on R². Yet, as we will see, such an object has no suitable limit as ǫ goes to 0. So, for future renormalization purposes, we rather consider:

g_ǫ(x)dx²=√

−lnǫ ǫ²e^2X^ǫ^(x)dx². 2.4.1 Volume form

The Riemannian volume on the manifold (R², g_ǫ) is given by:

Mǫ(dx) =√

−lnǫ ǫ²e^2X^ǫ^(x)dx, (2.9)

wheredxstands for the Lebesgue measure on R², and will be calledǫ-regularized critical measure.

The study of the limit of the random measures (M_ǫ(dx))_ǫ is carried out in [20,21] in a less general context. It is based on the study of the limit of the familyM_ǫ^′(dx) defined by:

M_ǫ^′(dx) := (2 ln1

ǫ −X_ǫ(x))e^2X^ǫ^(x)⁻^{2 ln}¹^ǫdx.

We will extend the results in [20] and prove

Theorem 2.1. Almost surely, the (locally signed) random measures(M_ǫ^′(dx))_ǫ>0 converge asǫ→0 towards a positive random measure M^′(dx) in the sense of weak convergence of measures. This limiting measure has full support and is atomless.

Concerning the Seneta-Heyde norming, we have

Theorem 2.2. Assume [A.1-5] and either [A.6] or [A.6’]. We have the convergence in probability in the sense of weak convergence of measures:

M_ǫ(dx)→ r2

πM^′(dx), as ǫ→0.

The proof of Theorem 2.2 is carried out in [21, section D] (in fact, it is assumed in [21] that the family of kernels (k(v, x, y))_v is translation invariant but adapting the proof is straightforward and thus left to the reader).

Beyond its conceptual importance, the Seneta-Heyde norming is crucial to establish, via Ka- hane’s convexity inequalities [34], the study of moments carried out in [20,21] and obtain

Proposition 2.3. Assume [A.1-5] and either [A.6] or [A.6’]. For each bounded Borel set A and q∈]− ∞,1[, the random variableM^′(A) possesses moments of order q. Furthermore, ifA has non trivial Lebesgue measure and x∈R²:

E[M^′(λA+x)^q]≃C(q, x)λ^ξ^M^′^(q) where ξ_M^′ is the power law spectrum of M^′:

∀q <1, ξ_M^′(q) = 4q−2q².

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Another important result about the modulus of continuity of the measure M^′ is established in [5]. We stress here that we pursue the discussion at a heuristic level since the result in [5] is not general enough to apply in our context (to be precise, it is valid for a well chosen family of kernels (k(v, x, y))_v in dimension 1 in order to get nice scaling relations for the associated measure M^′). Anyway, we expect this result to be true in greater generality and we will not use it in this paper: we are more interested in its conceptual significance. So, by analogy with [5], the measure M^′ is expected to possess the following modulus of continuity of ”square root of log” type: for all γ <1/2, there exists a random variable C almost surely finite such that

∀ball B ⊂B(0,1), M^′(B)6C(ln(1 +|B|⁻¹))⁻^γ. (2.10) Furthermore, the Hausdorff dimension of the carrier ofM^′ is 0. By analogy with the results that one gets in the context of multiplicative cascades [6], one also expects that the above theorem2.10 cannot be improved. In particular, the measureM^′does not possess a modulus of continuity better than a log unlike the subcritical situation explored in [27], where this property turned out to be crucial for the construction of the LBM as a whole Markov process. This remark is at the origin of the further complications arising in our paper (the critical case) in comparison with [27, 28] (the subcritical case).

2.4.2 Regularized Liouville Brownian motion

The main concern of this paper will be the Brownian motion associated with the metric tensor g_ǫ: following standard formulas of Riemannian geometry, one can associate to the Riemannian manifold (R², g_ǫ) a Brownian motion B^ǫ:

Definition 2.4 (ǫ-regularized critical Liouville Brownian motion, LBM_ǫ for short). For any fixed ǫ >0, we define the following diffusion on R². For any x∈R²,

B^ǫ,xt =x+ Z t

0

(−lnǫ)⁻^1/4ǫ⁻¹e⁻^X^ǫ⁽^B^ǫ,x^u ⁾dB¯u. (2.11) where B¯_t is a standard two-dimensional Brownian motion.

We stress that the fact that there is no drift term in the definition of the Brownian motion is typical from a scalar metric tensor in dimension 2. By using the Dambis-Schwarz Theorem, one can define the law of the LBMǫ as

Definition 2.5. For anyǫ >0 fixed andx∈R²,

B^ǫ,xt =x+B_hBǫ,xit, (2.12)

where(B_r)_r>0is a two-dimensional Brownian motion independent of the fieldXand thequadratic variationhB^ǫ,xi of B^ǫ,x is defined as

hB^ǫ,xit:= inf{s>0 : (−lnǫ)^1/2ǫ² Z s

0

e^2X^ǫ^(x+B^u⁾du>t}. (2.13) It will thus be useful to define the following quantity on R²×R₊:

F^ǫ(x, s) =ǫ² Z s

0

e^2X^ǫ^(x+B^u⁾du, (2.14)

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in such a way that the process hB^ǫ,xiis entirely characterized by:

(−lnǫ)^1/2F^ǫ(x,hB^ǫ,xit) =t. (2.15) Several standard facts can be deduced from the smoothness of Xǫ. For each fixed ǫ > 0, the LBM_ǫ a.s. induces a Feller diffusion on R², and thus a semi-group (P_t^ǫ)_t>0, which is symmetric w.r.t the volume formM_ǫ.

We will be mostly interested in establishing the convergence in law of the LBM_ǫ as ǫ → 0.

Basically, studying the convergence of the LBM_ǫ thus boils down to establishing the convergence of its quadratic variations hB^ǫ,xi.

3 Critical LBM starting from one fixed point

The first section is devoted to the convergence of the ǫ-regularized LBM when starting from one fixed point, sayx∈R². As in the case of the convergence of measures [20,21], the critical situation here is technically more complicated than in the subcritical case [27], though conceptually similar.

The first crucial step of the construction consists in establishing the convergence towards 0 of the family of functions (F^ǫ(x,·))_ǫ and then in computing the first order expansion of the maximum of the field X_ǫ along the Brownian path up to timet, and more precisely to prove that

smax∈[0,t]X_ǫ(x+B_s)−2 ln 1

ǫ → −∞, asǫ→0. (3.1)

This is mainly the content of subsection 3.2, after some preliminary lemmas in subsection 3.1.

Then our strategy will mainly be to adapt the ideas related to convergence of critical measures [20,21].

We further stress that, in the case of measures (see [20]), the content of subsection 3.2 is established thanks to comparison with multiplicative cascades measures and Kahane’s convexity inequalities. In our context, no equivalent result has been established in the context of multiplicative cascades in such a way that we have to carry out a direct proof.

3.1 Preliminary results about properties of Brownian paths

Let us consider a standard Brownian motion B on the plane R² starting at some given point x∈R². Let us consider the occupation measure µ_t of the Brownian motion up to time t >0 and let us define the function

∀ǫ∈]0,1], h(ǫ) = ln1

ǫln ln ln1

ǫ. (3.2)

The following result is proved in [46]

Theorem 3.1. There exists a deterministic constant c >0 such thatP^B-almost surely, the set E ={z∈R²; lim sup

ǫ→0

µt(B(z, ǫ)) ǫ²h(ǫ) =c} has full µ_t-measure.

We will need an extra elementary result about the structure of Brownian paths:

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Lemma 3.2. For everyp >2, we have almost surely:

Z

R²×R²

1

|x−y|²ln _|_x₋¹_y_|+ 2pµ_t(dx)µ_t(dy)<+∞. Proof of Lemma 3.2.We have:

E^Bh Z

R²×R²

1

|x−y|²ln _|_x₋¹_y_|+ 2pµ_t(dx)µ_t(dy)i

=E^Bh Z

[0,t]²

1

|B_r−B_s|²ln _|_B ¹

r−Bs|+ 2p drdsi

= Z

[0,t]²E^Bh 1

|r−s||B₁|²ln ¹

|r−s|^1/2|B1|+ 2p

idrds.

Let us compute for a 6 t the quantity E^B

h 1

a|B1|²ln ¹

a1/2|B1|+2p

i

. By using the density of the Gaussian law, we get:

E^B

h 1

a|B₁|²ln ¹

a^1/2|B1|+ 2p

i

= 1 2π

Z

R²

1 a|u|²ln ¹

a^1/2|u|+ 2pe⁻^|u|

2

2 du6

Z _∞

0

1 arln ¹

a^1/2r + 2pe⁻^r

2 2 dr

6 1 a

Z _∞

0

1

uln ¹_u+ 2pe⁻^u

2 2a du

6 1 a

Z a^1/4

0

1

uln _u¹ + 2pe⁻^u

2

2a du+1 a

Z t

a^1/4

1

uln ¹_u+ 2pe⁻^u

2

2a du+ 1 a

Z _∞

t

1

uln ¹_u+ 2pe⁻^u

2 2a du

6 1 a

Z a^1/4

0

1

uln^p _u¹du+1 a

Z t

0

1

uln _u¹ + 2pe⁻^2a¹^1/2 du+ 2 t²ln^p2e⁻^t

2 2a

6 4^p

a(p−1) ln^p⁻^{1 1}_a +C

ae⁻^2a¹^1/2 + 2 t²ln^p2e⁻^t

2 2a. Therefore

E^Bh Z

R²

Z

R²

1

|x−y|²ln _|_x₋¹_y_|+ 2pµ_t(dx)µ_t(dy)i 6C

Z t

0

Z t

0

1

|r−s|ln^p⁻¹ _|_r₋¹_s_| + 1

|r−s|e⁻

1

2|r−s|1/2 +e⁻ ^t

2 2|r−s|

drds.

As this latter quantity is obviously finite, the proof is complete.

3.2 First order expansion of the maximum of the field along Brownian paths To begin with, we claim:

Proposition 3.3. For allx∈R², almost surely inX, the family of random mapping t7→F^ǫ(x, t) converges to 0 in the space C(R₊,R₊) as ǫ→0.

Proof. Fixx∈R². Observe first that F^ǫ(x, t) =ǫ²

Z t

0

e^2X^ǫ^(x+B^r⁾dr= Z t

0

e^g^ǫ^(x+B^r⁾e^2X^ǫ^(x+B^r⁾⁻^2E^X^[X^ǫ^(x+B^r⁾²^]dr

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where

g_ǫ(u) = 2E^X[X_ǫ(u)²]−2 ln1 ǫ =

Z 1/ǫ 1

k(u, x, x)−1

u du.

By assumption [A.4], the function g_ǫ converges uniformly over the compact subsets of R². Fur- thermore, for each t >0, the set {x+B_s;s∈ [0, t]} is a compact set and g_ǫ converges uniformly over this compact set. So, even if it means considering Rt

0e^2X^ǫ^(x+B^r⁾⁻^2E^X^[X^ǫ^(x+B^r⁾²^]dr instead of F^ǫ(x, t), we may assume thatE^X[Xǫ(x+Br)²] = ln¹_ǫ, in which caseF^ǫ(x, t) is a martingale with respect to the filtration Fǫ =σ{X_r(x);ǫ6r, x ∈ R²}. As this martingale is nonnegative, it converges almost surely. We just have to prove that the limit is 0. To this purpose, we use a lemma in [23]. Translated into our context, it reads:

Lemma 3.4. The almost sure convergence of the family(F^ǫ(x, t))_ǫ towards0asǫ→0is equivalent to the fact that lim sup_ǫ_→₀F^ǫ(x, t) = +∞ under the probability measure defined by:

Q_|F_ǫ =t⁻¹F^ǫ(x, t)P^X.

The main idea of what follows is to prove that, under Q, F^ǫ(x, t) is stochastically bounded from below by the exponential of a Brownian motion so that lim sup_ǫ_→₀F^ǫ(x, t) = +∞ and we apply Lemma3.4 to conclude .

To carry out this argument, let us define a new probability measure Θ_ǫ on B(R²)⊗ Fǫ by Θ_ǫ_|B_(R²₎_⊗F_ǫ =t⁻¹e^2X^ǫ^(y)⁻^{2 ln}¹^ǫP^X(dω)µt(dy), (3.3) whereµ_tstands for the occupation measure of the Brownian motionB^xstarting fromx. We denote by EΘǫ the corresponding expectation. In fact, since the above definition defines a pre-measure on the ring B(R²)⊗S

ǫFǫ, one can define a measure Θ on B(R²)⊗ F by using Caratheodory’s extension theorem. We recover the relation Θ_|B_(A)_⊗F_ǫ = Θ_ǫ. Similarly, we construct the probability measure Qon F =σ S

ǫFǫ

by setting:

Q_|F_ǫ =t⁻¹F^ǫ(x, t)dP^X,

which is nothing but the marginal law of (ω, y)7→ω with respect to Θ_ǫ. We state a few elementary properties below. The conditional law ofy given Fǫ is given by:

Θ_ǫ(dy|Fǫ) = e^2X^ǫ^(y)⁻^{2 ln}¹^ǫ

F^ǫ(x, t) µ_t(dy).

IfY is aB(R²)⊗ Fǫ-measurable random variable then it has the following conditional expectation given Fǫ:

EΘǫ[Y|F^ǫ] = Z

R²

Y(y, ω)e^2X^ǫ^(y)⁻^{2 ln}¹^ǫ

F^ǫ(x, t) µt(dy).

Now we turn to the proof of Proposition3.3while keeping in mind this preliminary background.

Let us observe that it is enough to prove that the set {lim sup

ǫ→0

F^ǫ(x, t) = +∞}

has probability 1 conditionally toy under Θ to deduce that it satisfies Q {lim sup

ǫ→0

F^ǫ(x, t) = +∞}

= 1.

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So we have to compute the law of F^ǫ(x, t) under Θ(·|y). Recall the definition of h in (3.2). We have:

F^ǫ(x, t) = Z

R²

e^2X^ǫ^(u)⁻^{2 ln}¹^ǫµ_t(du) >

Z

R²

e^2X^ǫ^(u)⁻^{2 ln}¹^ǫ1B(y,ǫ)µ_t(du).

Let us now write

X_ǫ(u) =λ_ǫ(u, y)X_ǫ(y) +Z_ǫ(u, y) whereλ_ǫ(u, y) = ^K^ǫ^(u,y)

ln¹_ǫ andZ_ǫ(u, y) =X_ǫ(u)−λ_ǫ(u, y)X_ǫ(y). Observe that the process (Z_ǫ(u, y))_u_∈_R2

is independent of X_ǫ(y). Therefore F^ǫ(x, t)>

Z

R²

e^2X^ǫ^(u)⁻^{2 ln}¹^ǫ1B(y,ǫ)µ_t(du)

=e^2X^ǫ^(y)⁻^{2 ln}¹^ǫ Z

R²

e^2Z^ǫ^(u,y)+2(λ^ǫ^(u,y)⁻^1)X^ǫ^(y)1B(y,ǫ)µ_t(du)

=e^2X^ǫ^(y)⁻^{4 ln}¹^ǫ^+lnh(ǫ)× inf

u∈B(y,ǫ)e^2(λ^ǫ^(u,y)⁻^1)X^ǫ^(y)× 1 ǫ²h(ǫ)

Z

R²

e^2Z^ǫ^(u,y)1B(y,ǫ)µ_t(du).

Let us define

a_ǫ(y) = Z

R²

1B(y,ǫ)µ_t(du).

With the help of the Jensen inequality, we deduce F^ǫ(x, t)>e^2X^ǫ^(y)⁻^{4 ln}¹^ǫ^+ln^h(ǫ) inf

u∈B(y,ǫ)e^2(λ^ǫ^(u,y)⁻^1)X^ǫ^(y) a_ǫ(y)

ǫ²h(ǫ)exp 1 aǫ(y)

Z

B(y,ǫ)

2Z_ǫ(u, y)µ_t(du) . Let us set

Yǫ =aǫ(y)⁻¹ Z

R²

2Zǫ(u, y)1B(y,ǫ)µt(du).

Finally, for all R >0, we use the independence of Yǫ andXǫ(y) to get Θ

{lim sup

ǫ→0

F^ǫ(x, t) = +∞}|y

(3.4)

>Θ

{lim sup

ǫ→0

e^2X^ǫ^(y)⁻^{4 ln}¹^ǫ^+lnh(ǫ)× inf

u∈B(y,ǫ)e^2(λ^ǫ^(u,y)⁻^1)X^ǫ^(y)× aǫ(y)

ǫ²h(ǫ) ×exp(−R) = +∞}|y

×Θ(Y_ǫ > −R|y). (3.5)

Now we analyze the behaviour of each term in the above expression.

First, notice that Θ(Y_ǫ > −R|y) =P^X(Y_ǫ > −R) and thatY_ǫ is a centered Gaussian random variable underP^X with variance

a_ǫ(y)⁻² Z

B(y,ǫ)×B(y,ǫ)E^X

(X_ǫ(u)−λ_ǫ(u, y)X_ǫ(y))(X_ǫ(u^′)−λ_ǫ(u^′, y)X_ǫ(y))

µ_t(du)µ_t(du^′).

This quantity may be easily evaluated with assumption [A.2] and proved to be less than some constant C, which does not depend onǫand y∈ {x+B_s;s∈[0, t]}. Therefore

Θ(Y_ǫ > −R|y)>1−ρ(R) (3.6)

Liouville Brownian motion at criticality

HAL Id: hal-00914387

https://hal.archives-ouvertes.fr/hal-00914387

Liouville Brownian motion at criticality

Rémi Rhodes, Vincent Vargas

To cite this version:

Liouville Brownian motion at criticality

Contents

1 Introduction

2 Setup

3 Critical LBM starting from one fixed point